A holomorphic map into a Hilbert space with prescribed orthogonality

Question

This is a variation of my previous question.

Let $X\subset \mathbb{C}^n$ be a domain, and let $L:X\times X\to \mathbb{C}$ be such that $L(x,x)>0$, $L(y,x)=\overline{L(x,y)}$ and $L(\cdot,y)$ is holomorphic, for every $x,y$.

Does there always exist a holomorphic map $\lambda:X\to H\backslash\{0\}$, where $H$ is the Hilbert space, such that $\lambda(x)\bot \lambda(y)$ if and only if $L(x,y)=0$?